The metaplectic semigroup and related topics (Q1086687)

\textit{V. Bargmann} [Analytic Methods math. Phys., Conf. Bloomington, Indiana, 1968, 27-63 (1970; Zbl 0222.43008)] constructed projective representations, realized by integral operators on a reproducing kernel space, of the symplectic group \(G=Sp(2n, {\mathbb{R}})\) associated with the canonical commutation relations for a system with n degrees of freedom. The representation is unitary and was extended by Bargmann to a strongly continuous unitary representation of the universal covering group G. The author and the reviewer [Rep. Math. Phys. 17, 205-215 (1980; Zbl 0485.22017)] extended the representation from G to a semigroup S such that Sp(2n, \({\mathbb{R}})<S<Sp(2n, {\mathbb{C}})\). The author shows that S is connected and constructs the universal covering semigroup \(\tilde S.\) The representation of S is lifted to a strongly continuous representation of \(\tilde S.\) The representation operators are injective contractions and leave a subspace isomorphic to the Schwartz space invariant.